Equations &
Inequalities
Solving Linear
Equations
Unit 2
WHAT THIS UNIT IS
ABOUT
In this unit you will be learning
about the use of additive and
multiplicative inverses in solving
linear equations.
You will learn about the “number
machine” and how you can treat a
mathematical operation as a
“machine” that changes the
The Shipping department of a printing firm in Syria where
numbers you put in into different
copies of printed documents are sent out to customers.
numbers that come out. This is
similar to a copying machine in
that you put a certain number of master copies in. You press how many copies you want
and the correct number of pages comes out.
These techniques will help you to solve linear equations using the additive inverse and
the reciprocal or multiplicative inverse of numbers.
In this unit you will

Identify the multiplicative inverse or
reciprocal of different numbers.

Solve a linear equation by using the
additive inverse.

Solve a linear equation by using the
reciprocal or multiplicative inverse.

Solve for x in linear equations that
include addition and multiplication.

Identify the additive inverse or
reciprocal of different numbers.

Write an equation from a word
problem and solve the problem.
©PROTEC 2001
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Activity 1
Using multiplication number machines
A photocopier is a machine that changes a page or a document into a
number of pages or documents. The original page or document can be
called the input to the photocopying system. The copies are called the
output. In order to change the input into the output you need to enter the
number of copies you want. This number can be called the operation
In this activity you will learn how you can
use this ‘machines” approach to change
input numbers into output numbers.
1.1 Multiplication and the
photocopier.
If you press the number 5 and start on the
photocopier, the original page is translated into
5 copies
A
Draw up a table like the one opposite
and fill in the missing numbers
Number Machines are used
to show how input numbers
can be changed into output
numbers by a mathematical
operation.
Number
of
original
pages
6
The left column can be taken as the input
column. The middle column can be considered
to be the ‘operation’ and the last column the
output. An operation is the instruction given to
the machine.
The first row in the table shows that each of the
6 original pages in the first column is to be
‘operated’ on twice by the photocopier, giving
as output 12 copies.
2
2
3
1
Number
pressed
on the
copier
2
5
7
6
3
4
5
3
8
Number
of copies
20
6
16
21
1.2 Reversing the operation or the Multiplicative inverse or
Reciprocal
In some of the examples above you had to find the input from the output and the operation.
You did the reverse the operation.
Have another look at how you did this.
©PROTEC 2001
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A
Identify what you have to do to reverse the operations in the table
below. Then draw up your own table and fill it in.

Operation
Reverse
operation
Which is
equivalent
to?
2
3
4
5
6
7
8
9
10
 In reversing ‘multiply by 3’ we multiply
by ____.
 In reversing ‘multiply by 4’ we multiply
by ____.
 In reversing ‘multiply by 5’ we multiply
by ____.
 In reversing ‘multiply by 6’ we multiply
by ____.
 In reversing ‘multiply by 7’ we multiply
by ____.
To reverse
‘multiply by 2’
we ‘multiply by
½’
This means that ½ is a
multiplicative inverse
or reciprocal of 2.
B
In reversing ‘multiply by 2’ we multiply
by ____.
 In reversing ‘multiply by 8’ we multiply
by ____.
 In reversing ‘multiply by 9’ we multiply
by ____.
 In reversing ‘multiply by 10’ we multiply
by ____.
The reverse operation to multiplication is
division. Since dividing by a number is the
same as multiplying by its inverse, we say
that the reverse of multiplying by a number is
the same as multiplying by its (multiplicative)
inverse or reciprocal.
Use the information to fill in a table like the one below:
Number
Multiplicative inverse or
Reciprocal
3
4
5
6
7
8
©PROTEC 2001
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1.3 Using variables in Number Machines.
Suppose you want to photocopy x items by pressing the number 4 on the copier. What will
your output be? What must we multiply by to move from output to input? The same
principles apply even though you are dealing with variables instead of numbers. Some of the
examples in the table below use both variables and numbers.
A
Fill in the missing information
in the table.
1.4 Solving equations
involving single terms.
Input
x
Operation
y
a
3
5
6
7
8
Output
4x
3x
21b
24d
What do you understand by the expression
5x = 20
The statement that one quantity is equal to another is called an equation. If the
exponent of the variable involved is one, the equation is called linear equation.
You get 5x (as an output) by using the input x and the operation
5. The equation tells us that numerical value of the output
5x is 20. We want the numerical value of the input x.
A
Solve for the unknown
quantity or variable x in
the following table.
Input
x
x
x
x
Operation
5
6
7
8
Remember.
To keep an equation
balanced, whatever
you do to the output
you must also do its
numerical value.
Output
5x = 20
6x = 24
7x = 21
8x = 32
©PROTEC 2001
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Activity 2
Addition and its inverse
Buying & Selling
A man sells four cattle. He has 10 cattle
altogether. The “number machine” table
shows how the selling process changes the
amount of cattle the farmer had (input) into
the number she will have (output).
Input
10
Operation
-4
Output
6
2.1 Calves born & cattle sold
Each time calves are born the farmer
has more cattle. Each time they are
sold she has less.
A
Input
10
Explain in words what is
happening to the farmers
cattle in each row of the
“number machine” table
shown.
B
Now write equations for each row
with an empty cell. Use x to
represent the unknown.
C
Identify the rows in the above
table that involve reverse
operation.
D
Fill in the following table to
show how you could reverse the
operations shown.
Operation
-4
+5
10
-6
+2
-6
10
Output
6
7
7
18
10
Reverse operations.
We use reverse operation to refer to
undoing the transaction that has been
done, in other words, we start from
the output to the input.
Operation
-4
+5
-6
+7
-8
+9
-10
Reverse operation
A reverse operation is
called the additive
inverse of a forward
operation.
©PROTEC 2001
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2.2
A
Using a variable (x) in operations
Now complete the table
shown by replacing the
numbers in the input
column by a variable (x).
Input
x
Operation
-4
+5
-1
Output
x+5
x -1
x+10
x-3
x-6
x
x
-6
The output column
above is an algebraic
expression with the
variable x.
If you know what
the value of the
output is, then you
can make an
equation, e.g.
x - 4=5
2.3 Using the Additive inverse to solve
linear equations
Solving an equation using an input-operation-output model.
e.g. for the equation x-2 = 5
1.
To get to the output x-2, we used the input, x and the
operation, –2.
x-2
2.
x-2 = 5 means that the numerical value of the output x-2 is 5.
x-2=5
3.
To get the input x, we need to reverse the operation, –2 i.e.
add the additive inverse +2 to the output x-2.
x-2+2=5+2
4.
You must also do the same, add +2 to the other side of the
equation or the numerical value (5) to keep it balanced.
5.
Check this is true
x=7
LHS=x-2 = 7-2 = 5
RHS= 5
LHS=RHS
Remember: What you do to the output you must also do
its numerical
value. Output
Input
Operation
A Use the Additive inverses of the
x
-2
x-2 = 5
x
5
x+5=3
operations shown in the table to solve
x
-1
x
-1=3
the equations in the output columns.
x
10
x+10=15
x
-3
x-3= 4
x
-6
x-6=1
©PROTEC 2001
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Activity 3
Combining Addition and multiplication
So far you have dealt with multiplication and addition separately.


When a term consists of a variable and number separated by multiplication (or
division), we used the multiplicative inverse to separate them.
When a term consists of a variable and number separated by addition (or subtraction),
we used the additive inverse to separate them.
The activity below deals with equations involving both multiplication (or division) and
addition (or subtraction).
3.1
Using more than one “Number Machine” or operation.
A
Fill in the table by
calculating what
happens to the
input number as it
passes through
both operation 1
and operation 2.
B
Now show how you
would reverse the
process to move from
output to input.
Input
1
2
3
4
x
Operation 1
2
3
2
3
3
Operation 2
+1
-2
+3
-4
+2
Output
If variable and number are separated by + or -, use
additive inverse to reverse the process.
If variable and number are separated by  or , use
multiplicative inverse to reverse the process.
Remember BODMAS?
When you reverse
operations you must
use the reverse of this
order.
i.e. use SAMDOB
©PROTEC 2001
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3.2
A
B
Solving for x.
Complete the
equations in the
output column of
table, then solve for
x by reversing the
operations shown.
Input
x
x
x
x
x
Operation
1
3
2
2
3
3
Operation
2
-2
-1
+3
-4
+2
Output
3x-2 = 4
2x-1=5
?=9
?=8
?=7
Solve for x in the equations below using the additive inverses followed by
the multiplicative inverses or reciprocals.
1.
2.
3.
4.
5.
5x-2=18
3x+5=23
x-5=17
4x+10=2x+16
5x+3=-2x-4
3.3 Word problems & equations
Identify the input, operation and output in the following word problems and solve for x.
A
John’s age is three times that of James. John is two years older than
Mary. How old would::
 John and Mary be if James is 5 years old.
 James and John be if Mary is 10 years old.
 Mary and James be if John is 18 years old.
B
At a certain time of the day a tree’s shadow is twice its length. The tree
is 2m long. How long is the shadow?
C
I am thinking of a number. If I multiply my number by three and add 5 I
get 26. What is the number?
Activity 4
Solving
©PROTEC 2001
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